Final answer:
To find the area between the curves y = 3x²−3 and y = 2x+5, find the points of intersection, then integrate the difference of the curves over the interval between these points.
Step-by-step explanation:
To find the area between the graphs of y = 3x²−3 and y = 2x+5, we first need to find the points of intersection of the two curves. This is done by setting the two equations equal to each other: 3x²−3 = 2x+5. Solving this quadratic equation will give us the x-values where the curves intersect.
Once we have the points of intersection, we can integrate the difference of the functions over the interval defined by these points to find the area between the curves. Specifically, we will calculate the integral of (2x+5) - (3x²−3) with respect to x, from the smaller x-intersect to the larger x-intersect to get the total area encompassed.
Remember to review integral calculus and the properties of definite integrals when solving such problems, as they are essential in understanding how to calculate areas between curves.
To find the area between the two curves, we first need to find the points of intersection. Setting the two equations equal to each other and solving for x, we get 3x²−3 = 2x+5. This simplifies to 3x²-2x-8 = 0. Solving this quadratic equation, we find that x = -1 or x = 8/3.
Next, we integrate the difference of the two equations with respect to x from -1 to 8/3. The integral of (3x²−3) - (2x+5) is ∫(3x²-2x-8) dx.
Integrating term by term, we get (∫3x² dx) - (∫2x dx) - (∫8 dx). Evaluating the definite integral from -1 to 8/3, we find that the area between the two curves is approximately 51.83 square units.